Get Dietmar Planitzer's much-improved GLUT implementation, and use that. The GLUT that ships with MacOSX 10.1.x doesn't do game mode properly. This is a long-known bug that Apple refused to fix, but I believe Dietmar's version works properly.

You'll have to do a web search, or check the archives of the mac-opengl mailing list (http://lists.apple.com).

It's possible. I'm led to believe that GLUT uses Cocoa's GL implementation. You could send a message [NSOpenGLContext currentContext] and then do your stuff. You will still have to capture the display, change the resolution etc. I've posted Cocoa OpenGL sample code on my idisk (FÎanor's iDisk) which switches between fullscreen and windowed mode (called "Light Box.pbproj") and also cut and pasted the specific code in the MacGameWiki (search for Core Graphics).

In other news, I had thought that Apple had updated their GLUT and fixed this. If you're very intrepid you can edit their GLUT yourself ;-).